Theorem bnj873 32804

Description: Technical lemma for bnj69 32890. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj873.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj873.7	⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑)
bnj873.8	⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓)

Assertion

Ref	Expression
bnj873	⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)}

Distinct variable groups:   𝐷,𝑓,𝑔   𝑓,𝑛,𝑔   𝜑,𝑔   𝜓,𝑔

Allowed substitution hints:   𝜑(𝑓,𝑛)   𝜓(𝑓,𝑛)   𝐵(𝑓,𝑔,𝑛)   𝐷(𝑛)   𝜑′(𝑓,𝑔,𝑛)   𝜓′(𝑓,𝑔,𝑛)

Proof of Theorem bnj873

Step	Hyp	Ref	Expression
1		bnj873.4	. 2 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
2		nfv 1918	. . 3 ⊢ Ⅎ𝑔∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
3		nfcv 2906	. . . 4 ⊢ Ⅎ𝑓𝐷
4		nfv 1918	. . . . 5 ⊢ Ⅎ𝑓 𝑔 Fn 𝑛
5		bnj873.7	. . . . . 6 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑)
6		nfsbc1v 3731	. . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜑
7	5, 6	nfxfr 1856	. . . . 5 ⊢ Ⅎ𝑓𝜑′
8		bnj873.8	. . . . . 6 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓)
9		nfsbc1v 3731	. . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜓
10	8, 9	nfxfr 1856	. . . . 5 ⊢ Ⅎ𝑓𝜓′
11	4, 7, 10	nf3an 1905	. . . 4 ⊢ Ⅎ𝑓(𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)
12	3, 11	nfrex 3237	. . 3 ⊢ Ⅎ𝑓∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)
13		fneq1 6508	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛))
14		sbceq1a 3722	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ [𝑔 / 𝑓]𝜑))
15	14, 5	bitr4di 288	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ 𝜑′))
16		sbceq1a 3722	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ [𝑔 / 𝑓]𝜓))
17	16, 8	bitr4di 288	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ 𝜓′))
18	13, 15, 17	3anbi123d 1434	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)))
19	18	rexbidv 3225	. . 3 ⊢ (𝑓 = 𝑔 → (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)))
20	2, 12, 19	cbvabw 2813	. 2 ⊢ {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)}
21	1, 20	eqtri 2766	1 ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)}

Syntax hints:   ↔ wb 205   ∧ w3a 1085   = wceq 1539  {cab 2715  ∃wrex 3064  [wsbc 3711   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-fun 6420  df-fn 6421

This theorem is referenced by:  bnj849  32805  bnj893  32808